Quasinormal and Hyponormal Weighted Composition Operators on $$H^2$$ and $$A^2_{\alpha }$$Aα2 with Linear Fractional Compositional Symbol

In this paper, we study quasinormal and hyponormal composition operators \(W_{\psi ,\varphi }\)  with linear fractional compositional symbol \(\varphi \) on the Hardy and weighted Bergman spaces. We characterize the quasinormal composition operators induced on \(H^{2}\) and \(A_{\alpha }^{2}\) by these maps and many such weighted composition operators, showing that they are necessarily normal in all known cases. We eliminate several possibilities for hyponormal weighted composition operators but also give new examples of hyponormal weighted composition operators on \(H^2\) which are not quasinormal.     

Generalized Drazin invertibility and local spectral theory

If \(A\in B(\mathcal{X})\) is an upper triangular Banach space operator with diagonal \((A_1,A_2)\), \(A_1\) invertible and \(A_2\) quasinilpotent, then \(A_1^{-1}\oplus A_2\) satisfies either of the single-valued extension property, Dunford’s condition (C), Bishop’s property \((\beta )\), decomposition property \((\delta )\) or is decomposable if and only if \(A_1\) has the property. The operator \(A^{-1}_1\oplus 0\) is subscalar (resp., left polaroid, right polaroid) if and only if \(A_1\) is subscalar (resp., left polaroid, right polaroid). For Drazin invertible operators A, with Drazin inverse B, this implies that B satisfies any one of these properties if and only if A satisfies the property.     

Convergence of powers of composition operators on certain spaces of holomorphic functions defined on the right half plane

This paper studies the asymptotic behaviour of the powers \(C_\varphi ^n\) of a composition operator \(C_\varphi \) on certain spaces of holomorphic functions defined on the right half plane \(\mathbb {C}_+\). It is shown that for composition operators on the Hardy spaces and the standard weighted Bergman spaces, if the inducing map \(\varphi \) is not of parabolic type, then either the powers \(C_\varphi ^n\) converge uniformly only to 0 or they do not converge even strongly.     

Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let \(X_1\) and \(X_2\) be metric spaces equipped with doubling measures and let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\) respectively. We study multivariable spectral multipliers \(F(L_1, L_2)\) acting on the Cartesian product of \(X_1\) and \(X_2\). Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators \(L_1\) and \(L_2\), we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator \(F(L_1, L_2)\) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space \(X_1\times X_2\). We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

On the $$A_{\infty }$$ conditions for general bases

We discuss several characterizations of the \(A_\infty \) class of weights in the setting of general bases. Although they are equivalent for the usual Muckenhoupt weights, we show that they can give rise to different classes of weights for other bases. We also obtain new characterizations for the usual \(A_\infty \) weights.     

Weighted Harmonic Bloch Spaces on the Ball

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

Hyers–Ulam Stability of Differential Operators on Reproducing Kernel Function Spaces

In this paper, we investigate the Hyers–Ulam stability of the differential operators \(T_\lambda \) and D on the weighted Hardy spaces \(H_\beta ^2\) with the reproducing property. We obtain a necessary and sufficient condition in order that D is stable on \(H_\beta ^2\), and construct an example concerning the stability of \(T_\lambda \) on \(H_\beta ^2\). Moreover, we also investigate the Hyers–Ulam stability of the partial differential operators \(D_i\) on the several variables reproducing kernel space \(H_f^2(\mathbb {B}_d)\).     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals

We reduce the boundedness of operators in Morrey spaces \(L_p^r\left( {\mathbb R}^n\right) \), its preduals, \(H^{\varrho }L_p ({\mathbb R}^n)\), and their preduals \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) to the boundedness of the appropriate operators in Lebesgue spaces, \(L_p({\mathbb R}^n)\). Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.     

Uniform boundedness of Kantorovich operators in Morrey spaces

In this paper, the Kantorovich operators \(K_n, n\in \mathbb {N}\) are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference \(K_n(f)-f\) for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.     

Splines,lattice points,and arithmetic matroids

Let X be a \((d\times N)\)-matrix. We consider the variable polytope \(\varPi _X(u) = \{ w \ge 0 : X w = u \}\). It is known that the function \(T_X\) that assigns to a parameter \(u \in \mathbb {R}^d\) the volume of the polytope \(\varPi _X(u)\) is piecewise polynomial. The Brion–Vergne formula implies that the number of lattice points in \(\varPi _X(u)\) can be obtained by applying a certain differential operator to the function \(T_X\). In this article, we slightly improve the Brion–Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate \(T_X\)) and the space of nice differential operators (i.e. operators that leave \(T_X\) continuous). These two spaces are finite-dimensional homogeneous vector spaces, and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the \(\mathscr {P}\)-spaces studied by Ardila–Postnikov and Holtz–Ron in the context of zonotopal algebra and power ideals.     

Reverse Hölder Property for Strong Weights and General Measures

We present dimension-free reverse Hölder inequalities for strong \(A^*_p\) weights, \(1\le p < \infty \). We also provide a proof for the full range of local integrability of \(A_1^*\) weights. The common ingredient is a multidimensional version of Riesz’s “rising sun” lemma. Our results are valid for any nonnegative Radon measure with no atoms. For \(p=\infty \), we also provide a reverse Hölder inequality for certain product measures. As a corollary we derive mixed \(A_p^*-A_\infty ^*\) weighted estimates.     

Libera and Hilbert matrix operator on logarithmically weighted Bergman,Bloch and Hardy-Bloch spaces

We show that if α > 1, then the logarithmically weighted Bergman space \(A_{{{\log }^\alpha }}^2\) is mapped by the Libera operator L into the space \(A_{{{\log }^{\alpha - 1}}}^2\), while if α > 2 and 0 < ε ≤ α?2, then the Hilbert matrix operator H maps \(A_{{{\log }^\alpha }}^2\) into \(A_{{{\log }^{\alpha - 2 - \varepsilon }}}^2\).We show that the Libera operator L maps the logarithmically weighted Bloch space \({B_{{{\log }^\alpha }}}\), α ∈ R, into itself, while H maps \({B_{{{\log }^\alpha }}}\) into \({B_{{{\log }^{\alpha + 1}}}}\).In Pavlovi?’s paper (2016) it is shown that L maps the logarithmically weighted Hardy-Bloch space \(B_{{{\log }^\alpha }}^1\), α > 0, into \(B_{{{\log }^{\alpha - 1}}}^1\). We show that this result is sharp. We also show that H maps \(B_{{{\log }^\alpha }}^1\), α > 0, into \(B_{{{\log }^{\alpha - 1}}}^1\) and that this result is sharp also.     

Unitary Equivalence of Composition C$$^*$$-Algebras on the Hardy and Weighted Bergman Spaces

Let \(\varphi \) be an arbitrary linear-fractional self-map of the unit disk \({\mathbb {D}}\) and consider the composition operator \(C_{-1, \varphi }\) and the Toeplitz operator \(T_{-1,z}\) on the Hardy space \(H^2\) and the corresponding operators \(C_{\alpha , \varphi }\) and \(T_{\alpha , z}\) on the weighted Bergman spaces \(A^2_{\alpha }\) for \(\alpha >-1\). We prove that the unital C\(^*\)-algebra \(C^*(T_{\alpha , z}, C_{\alpha , \varphi })\) generated by \(T_{\alpha , z}\) and \(C_{\alpha , \varphi }\) is unitarily equivalent to \(C^*(T_{-1, z}, C_{-1, \varphi }),\) which extends a known result for automorphism-induced composition operators. For maps \(\varphi \) that are not automorphisms of \({\mathbb {D}}\), we show that \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })\) is unitarily equivalent to \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})\), where \({\mathcal {K}}_{\alpha }\) and \({\mathcal {K}}_{-1}\) denote the ideals of compact operators on \(A^2_{\alpha }\) and \(H^2\), respectively, and apply existing structure theorems for \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}\) to describe the structure of \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}\), up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps \(\varphi \) that fix a point on the unit circle.     

A gradient estimate on graphs with the $$\varvec{}$$-condition

Let \(\omega \) be an unbounded radial weight on \(\mathbb {C}^d\), \(d\ge 1\). Using results related to approximation of \(\omega \) by entire maps, we investigate Volterra type and weighted composition operators defined on the growth space \(\mathcal {A}^\omega (\mathbb {C}^d)\). Special attention is given to the operators defined on the growth Fock spaces.     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

On $$L^p$$-Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

We extended the known result that symbols from modulation spaces \(M^{\infty ,1}(\mathbb {R}^{2n})\), also known as the Sjöstrand’s class, produce bounded operators in \(L^2(\mathbb {R}^n)\), to general \(L^p\) boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from \(L^p\)-Sobolev spaces \(L^p_s(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) spaces with symbols from the modulation space \(M^{\infty ,1}(\mathbb {R}^{2n})\) are bounded, whenever \(s\ge n|1/p-1/2|.\) This estimate is sharp for all \(1< p<\infty \).     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.