Complex Analysis and Operator Theory最新文献

Within this article, the maximal function method is used to establish the Calderón-Zygmund estimates for the weak solutions of a class of non-uniformly elliptic equations

where (A(x,Du)approx |Du|^{p_1-2}+mu (x)|Du|^{p_2-2}), (F(x,f)approx |f|^{p_1-2}+mu (x)|f|^{p_2-2}) and (1<p_1<p_2), (0le mu (cdot )in C^{0,alpha }({mathbb {R}}^n),;alpha in (0,1]). The aforementioned problems arise as Euler-Lagrange equations for variational functionals that were originally presented and studied within the context of Homogenization and the Lavrentiev phenomenon by Marcellini (Arch Ration Mech Anal 105:267–284, 1989. https://doi.org/10.1007/BF00251503) and Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33–66, 1987. https://doi.org/10.1070/IM1987v029n01ABEH000958). They are distinctive in that they exhibit that the growth and ellipticity change between two distinct types of polynomial depending on the position. This feature is characteristic of strongly anisotropic materials. The contribution of this paper is closely tied to the significant advancements made by Colombo and Mingione (J Funct Anal 270:1416–1478, 2016. https://doi.org/10.1016/j.jfa.2015.06.022) in the qualitative analysis of double phase problems, as well as the related techniques used by Zhang et al. (Ann Polon Math, 114:45–65, 2015. https://doi.org/10.4064/ap114-1-4).

Let (alpha >-1) and assume that f is (alpha )-harmonic mapping defined in the unit disk that belongs to the Hardy class (h^p) with (pgeqslant 1). We obtain some sharp estimates of the type (|f(z)|le g(|r|) Vert f^*Vert _p) and (|Df(z)|le h(|r|)Vert f^*Vert _p). We also prove a Schwarz type lemma for the class of (alpha )-harmonic mappings of the unit disk onto itself fixing the origin.

We perform spectral analysis of a Sturm-Liouville operator on (L^2(mathbb {R}_+,dx)) which, through the Liouville transformation, is unitarily equivalent to the Schrödinger operator on (L^2(mathbb {R},dx)) with heavy negative potential (V(x)=-e^{2x}). This analysis clarifies some operator theory aspects of the setting of Fourier-Neumann expansions initiated by Varona in 1994.

In this paper, we study some characterizations of the Toeplitz and Hankel operators with positive operator-valued function as symbol on the vector-valued Fock-type spaces. We first discuss that the Bergman projection (P:L^p_{Psi }({mathcal {H}})rightarrow F^p_{Psi }({mathcal {H}})) is bounded for all (1le ple infty ), and obtain the duality of the vector-valued Fock-type spaces. Second, using operator-valued Carleson conditions, we give a complete characterization of the boundedness and compactness of the Toeplitz operators on (F^p_{Psi }({mathcal {H}})(1<p<infty )). Finally, we describe the boundedness (or compactness) of the Hankel operators (H_G) and (H_{G^*}) on (F_{Psi }^2({mathcal {H}})) in terms of a bounded (or vanishing) mean oscillation. We also give geometrical descriptions for the operator-valued spaces (BMO_Psi ^2) and (VMO_Psi ^2) defined in terms of the Berezin transform.

Given a bounded symmetric domain D, we study (positive) pluriharmonic functions on D and investigate a possible analogue of the family of Clark measures associated with a holomorphic function from D into the unit disc in (mathbb {C}).

Given a bounded operator Q on a Hilbert space (mathcal {H}), a pair of bounded operators ((T_1,T_2)) on (mathcal {H}) is said to be Q-commuting if one of the following holds:

We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for (Q=qI_{mathcal {H}}), where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions, which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where (nge 3). Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class.

This paper is the survey of some of our results related to q-deformations of the Fock spaces and related to q-convolutions for probability measures on the real line (mathbb {R}). The main idea is done by the combinatorics of moments of the measures and related q-cumulants of different types. The main and interesting q-convolutions are related to classical continuous (discrete) q-Hermite polynomial. Among them are classical ((q=1)) convolutions, the case (q=0), gives the free and Boolean relations, and the new class of q-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of q-convolutions. The main result is the construction of Brownian motion related to q-Discrete Hermite polynomial of type I.

In this paper, we extend the Voleterra integral operator (V_g) and its companion (J_g) to integral operator

Using a unified approach, we completely characterize the boundedness and compactness of (T_g^{n,m}) from one Fock space (F_alpha ^p) to another (F_beta ^q) for (0<p,qle infty ), (0<alpha ,beta <infty ). As a surprising case, we obtain that the boundedness (compactness) of (V_g) and (J_g) from (F_alpha ^p) to (F_beta ^q) is equivalent when the weight parameter (alpha <beta ). We also estimate the norms and essential norms of (T_g^{n,m}).

We consider a Schrödinger operator with random potential distributed according to a Poisson process. We show that under a uniform moment bound expectations of matrix elements of the resolvent as well as the integrated density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.

Let X be a complete measure space of finite measure. The Lebesgue transform of an integrable function f on X encodes the collection of all the mean-values of f on all measurable subsets of X of positive measure. In the problem of the differentiation of integrals, one seeks to recapture f from its Lebesgue transform. In previous work we showed that, in all known results, f may be recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection ({{,mathrm{{mathcal {A}}},}}) of all measurable subsets of X of positive measure. The first result of the present work is that the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of X. In the second result of this work we provide an independent argument that shows that the recourse to filters is a necessary consequence of the requirement that the process of recapturing f from its mean-values is associated to a natural transformation, in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals. In the Appendix we have proved, in a precise sense, that natural transformations fall within the general concept of homomorphism. As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of partial magma.