Sign-changing solutions for supercritical elliptic problems in domains with small holes

Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) ,

$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$

with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p _j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.

     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \({\phi, \psi: C\to A}\) are approximately unitarily equivalent if and only if

$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$

where T(A) is the tracial state space of A. In this paper we prove the following: Given \({\kappa\in KL(C,A)}\) with \({\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) and with κ([1 _C ]) = [1 _A ] and a continuous affine map \({\lambda: T(A)\to T_{\mathfrak f}(C)}\) which is compatible with κ, where \({T_{\mathfrak f}(C)}\) is the convex set of all faithful tracial states, there exists a unital monomorphism \({\phi: C\to A}\) such that

$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$

for all \({c\in C_{s.a.}}\) and \({\tau\in T(A).}\) Denote by \({{\rm Mon}_{au}^e(C,A)}\) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map

$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$

where KLT(C, A)⁺⁺ is the set of compatible pairs of elements in KL(C, A)⁺⁺ and continuous affine maps from T(A) to \({T_{\mathfrak f}(C).}\) Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \({\kappa\in KL(C(X), A)}\) with \({\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) for which there is no homomorphism h: C(X) → A so that [h] = κ.

     

Schatten-Herz Operators,Berezin Transform and Mixed Norm Spaces

For p, q > 0 we study operators T on the Bergman space \({A_{2}(\mathbb{D)}}\) in the disk such that \({\left(\sum_{j}\Vert T\Delta_{j}\Vert_{p}^{q}\right)^{1/q}<\infty,}\) where the norms \({\Vert\cdot\Vert_{p}}\) are in the Schatten class S _p (A ₂), the projection \({\Delta_{j}f=\sum_{n\in I_{j}}a_{n}z^{n}}\) for \({f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}\) and \({I_{j}=[2^{j}-1,2^{j+1} )\cap(\mathbb{N}\cup\{0\})}\) for \({j\in\mathbb{N}\cup\{0\}.}\) We consider the relation of this property with mixed norms of the Berezin transform of T and of the related function \({f_{T}(z)={\Vert}T(k_{z})\Vert}\) where k _z is the normalized Bergman kernel. These classes of operators denoted by S(p, q) are closely related when assumed to be positive with other sets of operators, like the class of positive operators on A ₂ for which \({\left(\sum_{j\geq0}(\sum_{n\in I_{j}}|\left\langle T^pe_{n},e_{n}\right\rangle |)^{q/p}\right)^{1/q}<\infty}\) , where \({\{e_{n}\}_{n\geq0}}\) is the canonical basis of A ₂; also we study the relation of Toeplitz operators in S(p, q) with the Schatten-Herz classes, where the decomposition is through dyadic annuli of the domain \({\mathbb{D}}\) .     

Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function \(\psi \in W^{m,p}({\mathbb{R}}^d)\) such that if \(\widehat{\psi}(0) = 1\) and ψ satisfies the Strang–Fix conditions to order m ? 1, then a scale averaged approximation formula holds for all \(f \in W^{m,p}({\mathbb{R}}^d)\) :

$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$

The dilations { a _j } are lacunary, for example a _j =  2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in \({W^{m - 1,p}({\mathbb{R}}^d)}\) the scale averaging is unnecessary and one has the simpler formula \(f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)\) . The Strang–Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or “spanning” criteria for the small scale affine system \(\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}\) in \(W^{m,p}({\mathbb{R}}^d)\) . We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

     

On commutator of generalized Aluthge transformations and Fuglede–Putnam theorem

Let \(A=U|A|\) be the polar decomposition of A on a complex Hilbert space \({\mathscr {H}}\) and \(0<s,t\). Then \({\widetilde{A}}_{s, t}=|A|^sU|A|^t\) and \({\widetilde{A}}_{s, t}^{(*)}=|A^*|^sU|A^*|^t\) are called the generalized Aluthge transformation and generalized \(*\)-Aluthge transformation of A, respectively. A pair (A, B) of operators is said to have the Fuglede–Putnam property (breifly, the FP-property) if \(AX=XB\) implies \(A^*X=XB^*\) for every operator X. We prove that if (A, B) has the FP-property, then \(({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})\) and \((({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})\) has the FP-property for every \(s,t>0\) with \(s+t=1\). Also, we prove that \(({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})\) has the FP-property if and only if \((({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})\) has the FP-property, where A, B are invertible and \( 0 < s, t \) with \( s + t =1\). Moreover, we prove that if \(0 < s, t\) and \({\widetilde{A}}_{s, t}\) is positive and invertible, then \(\left\| {\widetilde{A}}_{s, t}X-X{\widetilde{A}}_{s, t}\right\| \le \left\| A\right\| ^{2t}\left\| ({\widetilde{A}}_{s, t})^{-1}\right\| \left\| X\right\| \) for every operator X. Also, if \( 0 <s, t\) and X is positive, then \(\left\| |{\widetilde{A}}_{s, t}|^{2r} X-X|{\widetilde{A}}_{s, t}|^{2r}\right\| \le \frac{1}{2}\left\| |A|\right\| ^{2r}\left\| X\right\| \) for every \(r>0\).     

Stability of volume comparison for complex convex bodies

We prove the stability of the affirmative part of the solution to the complex Busemann–Petty problem. Namely, if K and L are origin-symmetric convex bodies in \({{\mathbb C}^n}\), n = 2 or n = 3, \({\varepsilon >0 }\) and \({{\rm Vol}_{2n-2}(K\cap H) \le {\rm Vol}_{2n-2}(L \cap H) + \varepsilon}\) for any complex hyperplane H in \({{\mathbb C}^n}\) , then \({({\rm Vol}_{2n}(K))^{\frac{n-1}n}\le({\rm Vol}_{2n}(L))^{\frac{n-1}n} + \varepsilon}\) , where Vol_2n is the volume in \({{\mathbb C}^n}\) , which is identified with \({{\mathbb R}^{2n}}\) in the natural way.     

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation

$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 ? (α/N) < p < (2N ? α)/(N?2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$

where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

     

A Unified Framework for Linear Dimensionality Reduction in L1

For a family of interpolation norms \({\| \cdot \|_{1,2,s}}\) on \({\mathbb{R}^{n}}\), we provide a distribution over random matrices \({\Phi_s \in \mathbb{R}^{m \times n}}\) parametrized by sparsity level s such that for a fixed set X of K points in \({\mathbb{R}^{n}}\), if \({m \geq C s \log(K)}\) then with high probability, \({\frac{1}{2}\| \varvec{x} \|_{1,2,s} \leq \| \Phi_s (\varvec{x}) \|_1 \leq 2 \| \varvec{x} \|_{1,2,s}}\) for all \({\varvec{x} \in X}\). Several existing results in the literature roughly reduce to special cases of this result at different values of s: For s = n, \({\| \varvec{x} \|_{1,2,n}\equiv \| \varvec{x} \|_{1}}\) and we recover that dimension reducing linear maps can preserve the ?₁-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For s = 1, \({\| \varvec{x} \|_{1,2,1}\equiv \| \varvec{x} \|_{2}}\), and we recover an ?₂/?₁ variant of the Johnson–Lindenstrauss Lemma for Gaussian random matrices. Finally, if \({\varvec{x}}\) is s- sparse, then \({\| \varvec{x} \|_{1,2,s} = \| \varvec{x} \|_1}\) and we recover that s-sparse vectors in \({\ell_1^n}\) embed into \({\ell_1^{\mathcal{O}(s \log(n))}}\) via sparse random matrix constructions.     

On random arithmetical functions. I

Let A _Q be the group of complex unit roots of an integer order \( Q \geqslant 2 \). Let \( {\xi_p}\left( {p \in \mathcal{P}} \right) \) be independent random variables distributed uniformly on the set A _Q , where \( \mathcal{P} \)is the set of primes. Let f be a completely multiplicative function defined on \( \mathcal{P} \) by f(p) = ξ _p . We investigate the summatory function of f(n) and the density of those n for which f(n + j) = κ _j (j = 0, …, t), where κ _j ∈ A _Q .     

New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access network for 2-dimensional image transmission. There is a one-to-one correspondence between an \((m, n, w, \lambda )\)-OOSPC and a \((\lambda +1)\)-(mn, w, 1) packing design admitting an automorphism group isomorphic to \(\mathbb {Z}_m\times \mathbb {Z}_n\). In 2010, Sawa gave a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) which contains a unique element of order 2. In this paper, we study the existence of one-factor of Köhler graph of \(\mathbb {Z}_m\times \mathbb {Z}_n\) having three elements of order 2. It is proved that there is a one-factor in the Köhler graph of \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\) relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where \(p\equiv 5\pmod {12}\) is a prime and \(1\le \epsilon ,\epsilon '\le 2\). Using this one-factor, we construct a strictly \(\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}\)-invariant regular \(G^*(p,2^{\epsilon +\epsilon '},4,3)\) relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant regular G-designs, we construct more strictly \(\mathbb {Z}_{m}\times \mathbb {Z}_{n}\)-invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal \((2^{\epsilon }m,2^{\epsilon '}n,4,2)\)-OOSPC for any \(\epsilon ,\epsilon '\in \{0,1,2\}\) with \(\epsilon +\epsilon '>0\) and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set \(\{p\equiv 5\pmod {12}:p\) is a prime, \(p<\)1,500,000}.     

On ℤ<Subscript>2</Subscript>-graded identities of the super tensor product of <Emphasis Type="Italic">UT</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">F</Emphasis>) by the Grassmann algebra

Let F be a field of characteristic zero and E be the unitary Grassmann algebra generated over an infinite-dimensional F-vector space L. Denote by \(\mathcal{E} = \mathcal{E}^{(0)} \oplus \mathcal{E}^{(1)}\) an arbitrary ?₂-grading of E such that the subspace L is homogeneous. Given a superalgebra A = A ⁽⁰⁾ ⊕ A ⁽¹⁾, define the superalgebra \(A\hat \otimes \mathcal{E}\) by \(A\hat \otimes \mathcal{E} = (A^{(0)} \otimes \mathcal{E}^{(0)} ) \oplus (A^{(1)} \otimes \mathcal{E}^{(1)} )\). Note that when E is the canonical grading of E then \(A\hat \otimes \mathcal{E}\) is the Grassmann envelope of A. In this work we find bases of ?₂-graded identities and we describe the ?₂-graded codimension and cocharacter sequences for the superalgebras \(UT_2 (F)\hat \otimes \mathcal{E}\), when the algebra UT ₂(F) of 2 ×2 upper triangular matrices over F is endowed with its canonical grading.     

Stability of some versions of the Prékopa–Leindler inequality

We study the behavior of positive solutions of the following Dirichlet problem

$$\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}\right. $$

when s → p ^?. Here \({p >1 , s\,{\in}\,]1,p]}\) and q > p with \({q\leq\frac{Np}{N-p}}\) if N > p.

     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).     

Stabilization of a Class of Uncertain Systems

We consider the problem to synthesize a stabilizing control u synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where A ∈ ?^n×n and B ∈ ?^n×m, while the elements α_i,j(·) of the matrix A are uniformly bounded nonanticipatory functionals of arbitrary nature. If the system is continuous, then the elements of the matrix B are continuous and uniformly bounded functionals as well. If the system is pulse-modulated, then the elements of the matrix B are differentiable uniformly bounded functions of time. It is assumed that k isolated uniformly bounded elements \({\alpha _{{i_l},{j_l}}}\left( \cdot \right)\) satisfying the condition \(\mathop {\inf }\limits_{\left( \cdot \right)} \left| {{\alpha _{{i_l},{j_l}}}\left( \cdot \right)} \right|{\alpha _ - } > 0,\quad l \in \overline {1,k}\) are located above the main diagonal of the matrix A(·), where G _k is the set of all isolated elements of the system, J₁ is the set of indices of rows of matrix A(·) containing isolated elements, and J₂ is the set of indices of its rows free of isolated elements. It is assumed that other elements located above the main diagonal are sufficiently small provided that their row indices belong to J₁, i.e., \(\mathop {\sup }\limits_{\left( \cdot \right)} \left| {{\alpha _{i,j}}\left( \cdot \right)} \right| < \delta ,\quad {\alpha _{i,j}} \notin {G_k},\quad i \in {J_1},\quad j > i\). All other elements located above the main diagonal are uniformly bounded. The relation u = S(·)x is satisfied in the continuous case, while the relation u = ξ(t) is satisfied in the pulse-modulated case; here the components of the vector ξ are outputs of synchronous pulse elements. Constructing a special quadratic Lyapunov function, one can determine a matrix S(·) such that the closed system becomes globally exponentially stable in the continuous case. In the pulse-modulated case, input pulses are synthesized such that the system becomes globally asymptotically stable.     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

On the exceptional sets in Sylvester expansions*

For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d _j (x),?j?≥?1} is a sequence of positive integers satisfying d₁(x)?≥?2 and d_j?+?1(x)?≥?d _j (x)(d _j (x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?⁺ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set

$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$

and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)     

Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields

$$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$

\( n_1, n_2 \ne 0\). We show that the operator

$$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$

well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).

     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.