Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case

We study necessary and sufficient conditions for embeddings of Besov and Triebel-Lizorkin spaces of generalized smoothness B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), respectively, into generalized H?lder spaces L_￥,r^m(·)( \mathbb Rⁿ)\Lambda_{\infty,r}^{\mu(\cdot)}(\ensuremath {\ensuremath {\mathbb {R}}^{n}}). In particular, we are able to characterize optimal embeddings for this class of spaces provided q>1. These results improve the embedding assertions given by the continuity envelopes of B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), which were obtained recently solving an open problem of D.D. Haroske in the classical setting.     

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

The Boundedness of Monge-Ampère Singular Integral Operators

We first define molecules for Hardy spaces H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}) associated with a family F\mathcal{F} of sections which is closely related to the Monge-Ampère equation and prove their molecular characters. As an application, we show that Monge-Ampère singular operators are bounded on H¹_F(\mathbbRⁿ)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}).     

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

In this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of the L_r^2p(\mathbbR^d;\mathbbR¹)×L_r²(\mathbbR^d;\mathbbR^d)L_{\rho}^{2p}({\mathbb{R}^{d}};{\mathbb{R}^{1}})\times L_{\rho}^{2}({\mathbb{R}^{d}};{\mathbb{R}^{d}}) valued solution of backward stochastic differential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.     

On Singular Integral Operators with Rough Kernel Along Surface

In this note we give a simple method to transfer the effect of the surface to the radial function in the kernel of singular integral along surface. Using this idea, we give some continuity of the singular integrals along surface with Hardy space function kernels on some function spaces, such as L^p(\mathbb Rⁿ),L^p(\mathbb Rⁿ,w){L^p({\mathbb R}^n),L^p({\mathbb R}^n,\omega)}, Triebel–Lizorkin spaces [(F)\dot]_p^s,q(\mathbb Rⁿ){{\dot F}_{p}^{s,q}({\mathbb R}^n)}, Besov spaces [(B)\dot]_p^s,q(\mathbb Rⁿ){{\dot B}_{p}^{s,q}({\mathbb R}^n)}, generalized Morrey spaces L^p,f(\mathbb Rⁿ){L^{p,\phi}({\mathbb R}^n)} and Herz spaces [(K)\dot]_p^{a, q}(\mathbb Rⁿ){\dot K_p^{\alpha, q}({\mathbb R}^n)}. Our results improve and extend substantially some known results on the singular integral operators along surface.     

On the means of Fourier integrals and Bernstein inequality in the two-weighted setting

The norm estimation problem for Cesaro and Abel–Poisson operators acting from L_w^p(\mathbbR){L_{w}^{p}(\mathbb{R})} to L_v^q (\mathbbR){L_{v}^{q} (\mathbb{R})} where 1 < p ≤ q < ∞ was investigated. These results were generalized to the multidimensional case and applied to obtain generalizations of the Bernstein inequality for integral functions of finite degree of one and several variables.     

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      8. Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}      Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368 We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.      9. Density of the polynomials in Hardy and Bergman spaces of slit domains      John R. Akeroyd 《Arkiv f?r Matematik》2011,49(1):1-16 It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbAt(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in Ht(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.      10. The coherence of complemented ideals in the space of real analytic functions      Paweł Domański  Dietmar Vogt 《Mathematische Annalen》2010,347(2):395-409 We characterize when an ideal of the algebra ${A(\mathbb{R}^d)}We characterize when an ideal of the algebra A(\mathbbRd){A(\mathbb{R}^d)} of real analytic functions on \mathbbRd{\mathbb{R}^d} which is determined by the germ at \mathbb Rd{\mathbb {R}^d} of a complex analytic set V is complemented under the assumption that either V is homogeneous or V?\mathbbRd{V\cap \mathbb{R}^d} is compact. The characterization is given in terms of properties of the real singularities of V. In particular, for an arbitrary complex analytic variety V complementedness of the corresponding ideal in A(\mathbbRd){A(\mathbb{R}^d)} implies that the real part of V is coherent. We also describe the closed ideals of A(\mathbbRd){A(\mathbb{R}^d)} as sections of coherent sheaves.      11. Equidistribution of expanding measures with local maximal dimension and diophantine approximation      Ronggang Shi 《Monatshefte für Mathematik》2012,149(1):513-541 We consider improvements of Dirichlet’s Theorem on the space of matrices Mm,n(\mathbb R){M_{m,n}(\mathbb R)}. It is shown that for a certain class of fractals K ì [0,1]mn ì Mm,n(\mathbb R){K\subset [0,1]^{mn}\subset M_{m,n}(\mathbb R)} of local maximal dimension Dirichlet’s Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.      12. End-point maximal regularity and its application to two-dimensional Keller–Segel system      Takayoshi Ogawa  Senjo Shimizu 《Mathematische Zeitschrift》2010,264(3):601-628 We prove the maximal L ρ regularity of the Cauchy problem of the heat equation in the Besov space [(B)\dot]1,r0(\mathbbRn){\dot{B}_{1,\rho}^0(\mathbb{R}^n)}, 1 < ρ ≤ ∞, which is not UMD space. And as its application, we establish the time local well-posedness of the solution of two dimensional nonlinear parabolic system with the Poisson equation in [(B)\dot]1,20(\mathbbR2){\dot{B}_{1,2}^0(\mathbb{R}^2)} , where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time. The small data global existence is also obtained in the same class.      13. The affine preservers of non-singular matrices      Clément de Seguins Pazzis 《Archiv der Mathematik》2010,95(4):333-342 When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of Mn(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GLn(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of Mp,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M2(\mathbbF2){{\rm M}_2(\mathbb{F}_2)} that stabilize GL2(\mathbbF2){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF2{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp4(\mathbbF2){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS6{\mathfrak{S}_6}.      14. Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves      Vladimir Rabinovich  Stefan Samko 《Integral Equations and Operator Theory》2011,69(3):405-444 The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces Lp(·)(\mathbbR +,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR+{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.      15. Spectral Approximation and Index for Convolution Type Operators on Cones on L^{p}(\mathbb {R}^2)      H. Mascarenhas  B. Silbermann 《Integral Equations and Operator Theory》2009,65(3):415-448 We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on Lp(\mathbb R2)L^{p}(\mathbb {R}^2), with 1 < p < ∞. To each operator sequence (An) we associate a family of operators Wx(An) ? L(Lp(\mathbb R2))W_{x}(A_{n}) \in \mathcal {L}(L^{p}(\mathbb {R}^2)) parametrized by x in some index set. When all Wx(An) are Fredholm, the so-called approximation numbers of An have the α-splitting property with α being the sum of the kernel dimensions of Wx(An). Moreover, the sum of the indices of Wx(An) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the e\epsilon-pseudospectrum, norms of inverses and condition numbers are also obtained.      16. The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework      Matthias Hieber  Yoshihiro Shibata 《Mathematische Zeitschrift》2010,265(2):481-491 Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u 0 is small enough in the H\frac12s(\mathbbR3){H^{\frac12}_{\sigma}(\mathbb{R}^3)} -norm.      17. Triple Hilbert Transforms Along Polynomial Surfaces      Yong-Kum Cho  Sunggeum Hong  Joonil Kim  Chan Woo Yang 《Integral Equations and Operator Theory》2009,65(4):485-528 Given W ì \mathbbZ+3\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in Lp(\mathbbR4)L^p({\mathbb{R}}^4).      18. Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system      Jihong Zhao  Qiao Liu  Shangbin Cui 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):1-18 In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L\fracn2(\mathbbRn){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K1(K2|b|)|b|t-\frac|b|2-1+\fracn2q{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K 1 and K 2. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]-2+\fracnpp,￥(\mathbbRn){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).      19. Congruence-simple subsemirings of ℚ+      Vítězslav Kala  Miroslav Korbelář 《Semigroup Forum》2010,81(2):286-296 Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ+. Even the class CongSimp(\mathbb Q+)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ+ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ+. We classify all the maximal elements of CongSimp(\mathbbQ+)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ+)\{\mathbbQ+}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.      20. Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms      Ilham A. Aliev 《Integral Equations and Operator Theory》2009,65(2):151-167 We introduce new potential type operators Jab = (E+(-D)b/2)-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with Jαβ. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces Hab, p(\mathbbRn)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

The coherence of complemented ideals in the space of real analytic functions

We characterize when an ideal of the algebra ${A(\mathbb{R}^d)}We characterize when an ideal of the algebra A(\mathbbR^d){A(\mathbb{R}^d)} of real analytic functions on \mathbbR^d{\mathbb{R}^d} which is determined by the germ at \mathbb R^d{\mathbb {R}^d} of a complex analytic set V is complemented under the assumption that either V is homogeneous or V?\mathbbR^d{V\cap \mathbb{R}^d} is compact. The characterization is given in terms of properties of the real singularities of V. In particular, for an arbitrary complex analytic variety V complementedness of the corresponding ideal in A(\mathbbR^d){A(\mathbb{R}^d)} implies that the real part of V is coherent. We also describe the closed ideals of A(\mathbbR^d){A(\mathbb{R}^d)} as sections of coherent sheaves.     

Equidistribution of expanding measures with local maximal dimension and diophantine approximation

We consider improvements of Dirichlet’s Theorem on the space of matrices M_m,n(\mathbb R){M_{m,n}(\mathbb R)}. It is shown that for a certain class of fractals K ì [0,1]^mn ì M_m,n(\mathbb R){K\subset [0,1]^{mn}\subset M_{m,n}(\mathbb R)} of local maximal dimension Dirichlet’s Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.     

End-point maximal regularity and its application to two-dimensional Keller–Segel system

We prove the maximal L ^ρ regularity of the Cauchy problem of the heat equation in the Besov space [(B)\dot]_1,r⁰(\mathbbRⁿ){\dot{B}_{1,\rho}^0(\mathbb{R}^n)}, 1 < ρ ≤ ∞, which is not UMD space. And as its application, we establish the time local well-posedness of the solution of two dimensional nonlinear parabolic system with the Poisson equation in [(B)\dot]_1,2⁰(\mathbbR²){\dot{B}_{1,2}^0(\mathbb{R}^2)} , where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time. The small data global existence is also obtained in the same class.     

The affine preservers of non-singular matrices

When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of M_n(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GL_n(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of M_p,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M₂(\mathbbF₂){{\rm M}_2(\mathbb{F}_2)} that stabilize GL₂(\mathbbF₂){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF₂{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp₄(\mathbbF₂){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS₆{\mathfrak{S}_6}.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

Spectral Approximation and Index for Convolution Type Operators on Cones on L^{p}(\mathbb {R}^2)

We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on L^p(\mathbb R²)L^{p}(\mathbb {R}^2), with 1 < p < ∞. To each operator sequence (A_n) we associate a family of operators W_x(A_n) ? L(L^p(\mathbb R²))W_{x}(A_{n}) \in \mathcal {L}(L^{p}(\mathbb {R}^2)) parametrized by x in some index set. When all W_x(A_n) are Fredholm, the so-called approximation numbers of A_n have the α-splitting property with α being the sum of the kernel dimensions of W_x(A_n). Moreover, the sum of the indices of W_x(A_n) is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the e\epsilon-pseudospectrum, norms of inverses and condition numbers are also obtained.     

The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u ₀ is small enough in the H^\frac12_s(\mathbbR³){H^{\frac12}_{\sigma}(\mathbb{R}^3)} -norm.     

Triple Hilbert Transforms Along Polynomial Surfaces

Given W ì \mathbbZ₊³\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in L^p(\mathbbR⁴)L^p({\mathbb{R}}^4).     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

Congruence-simple subsemirings of ℚ+

Commutative congruence-simple semirings have already been characterized with the exception of the subsemirings of ℝ⁺. Even the class CongSimp(\mathbb Q⁺)\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb {Q}^{+}) of all congruence-simple subsemirings of ℚ⁺ has not been classified yet. We introduce a new large class of the congruence-simple saturated subsemirings of ℚ⁺. We classify all the maximal elements of CongSimp(\mathbbQ⁺)\mathit{\mathcal{C}ong\mathcal {S}imp}(\mathbb{Q}^{+}) and show that every element of CongSimp(\mathbbQ⁺)\{\mathbbQ⁺}\mathit{\mathcal{C}ong\mathcal{S}imp}(\mathbb{Q}^{+})\setminus\{\mathbb{Q}^{+}\} is contained in at least one of them.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).