Volume 25, Issue 2

In this paper we study the approximation of functions from generalized Morrey spaces by nice functions. We introduce a new subspace whose elements can be approximated by infinitely differentiable compactly supported functions. This provides, in particular, an explicit description of the closure of the set of such functions in generalized Morrey spaces.

The paper is concerned with the Cauchy problem for a nonlinear generalized heat equation which is related to the generalized Gauss–Weierstrass semigroup via Duhamel’s principle. For the initial data we assume that they belong to some fractional Sobolev spaces. We study the existence and uniqueness of mild and strong solutions which are local in time. Moreover, they are smooth functions and belong to Lebesgue spaces with respect to the space variable. We use both fixed point arguments and mapping properties of the generalized Gauss–Weierstrass semigroup. Finally, we study the well-posedness of the problem.

This paper presents a new principle related to an arbitrary meromorphic function w in a given domain D . The main component of this principle gives (first time) lower bounds for |w′|{|w^{\prime}|} for a similar general class of functions. The principle can qualitatively be stated as follows: any set of simple a -points of w contains a “large” subset of complex values, where we have lower bounds for |w′|{|w^{\prime}|} and upper bounds for |w(h)|{|w^{(h)}|}, h>1{h>1}.

The value distribution and, in particular, the numbers of a -points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a -points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a -points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a -points and Ahlfors islands of similar solutions.

Some related questions concerning the measurability properties of real-valued functions with respect to a certain class of measures are discussed.

In this note we prove a modular variable Orlicz inequality for the local maximal operator. This result generalizes several Orlicz and variable exponent modular inequalities that have appeared previously in the literature.

In this note we prove a one-dimensional Rellich inequality in variable exponent Lebesgue spaces, with exponents that satisfy a condition of log-Hölder type.

We present a certain generalization of Smirnov’s theorem on functions from the Hardy spaces Hp{H_{p}}. We provide some applications of the proposed generalization. Namely, we give an equivalent characterization of outer analytic rectangular matrix functions, and give a simple proof of the uniqueness of spectral factorization of rank deficient matrices.

We study the variation of Neumann eigenvalues of the p -Laplace operator under quasiconformal perturbations of space domains. This study allows us to obtain the lower estimates of Neumann eigenvalues in fractal type domains. The proposed approach is based on the geometric theory of composition operators in connection with the quasiconformal mapping theory.

In this paper, we study the Riesz potential ( G -Riesz potential) generated by the Gegenbauer differential operator Gλ=(x2-1)12-λ⁢dd⁢x⁢(x2-1)λ+12⁢dd⁢x,x∈(1,∞),λ∈(0,12).G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G -Riesz potential IGα{I_{G}^{\alpha}}, 0<α<2⁢λ+1{0<\alpha<2\lambda+1}, is bounded from the G -Morrey space Lp,λ,γ{L_{p,\lambda,\gamma}} to Lq,λ,γ{L_{q,\lambda,\gamma}} if and only if 1p-1q=α2⁢λ+1-γ,1<p<2⁢λ+1-γα.\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1<p<\frac{2% \lambda+1-\gamma}{\alpha}. Also, we prove that the G -Riesz potential IGα{I_{G}^{\alpha}} is bounded from the G -Morrey space L1,λ,γ{L_{1,\lambda,\gamma}} to the weak G -Morrey space W⁢Lq,λ,γ{WL_{q,\lambda,\gamma}} if and only if 1-1q=α2⁢λ+1-γ.1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

For a given parameterization of a Jordan curve, we define the notion of summability or classes of measurable functions on a contour where a new integral is introduced. It is shown that natural functional spaces defining summability for non-rectifiable Jordan curves are the Lebesgue spaces with the weighted norm. For non-rectifiable Jordan curves where an integral was previously defined for continuous (Hölder) functions [Y. Guseynov, Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 103], a weight function is constructed which, in general, is not summable by parameter, and a weighted functional space (summability) is defined where the new integral exists.

In this paper, we define and investigate fractional cosine and sine transforms for functions as well as distributions. For these transforms, several properties are provided, the representation theorems are proved and the analyticity is established.

Following the ideas of our previous research, in this paper we continue the study of new Bergman-type spaces on the unit disc with mixed norm in terms of Fourier coefficients. Here we deal with the case where the sequence of norms of Fourier coefficients in the Orlicz–Morrey space in radial variable belongs to lq{l^{q}}. We study the boundedness of the Bergman projection and provide a description of functions in these spaces via the behavior of their Taylor coefficients.

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function f on ℝn{{\mathbb{R}}^{n}} is obtained under the assumption that f belongs to Lp{L^{p}}. The explicit value of the coefficient is found for the cases p=1{p=1} and p=2{p=2}.

We introduce grand Lebesgue sequence spaces and study various operators of harmonic analysis in these spaces, e.g., maximal, convolution, Hardy, Hilbert, and fractional operators, among others. Special attention is paid to fractional calculus, including the density of the discrete version of a Lizorkin sequence test space in vanishing grand spaces.

In this paper, we aim to deal with the boundedness and the weak-type boundedness for the generalized fractional integral operators on generalized Orlicz–Morrey spaces of the second kind over non-doubling metric measure spaces, as an extension of [Y. Sawano and T. Shimomura, Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces, Z. Anal. Anwend. 36 2017, 2, 159–190], [Y. Sawano and T. Shimomura, Generalized fractional integral operators over non-doubling metric measure spaces, Integral Transforms Spec. Funct. 28 2017, 7, 534–546] and [I. Sihwaningrum, H. Gunawan and E. Nakai, Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces, Math. Nachr., to appear].

A trigonometric sine series with special properties is constructed. One of the properties of this series shows that Bary’s conjecture on subsequences of partial sums of a sine series is not valid.