Yoshihiro Mizuta

Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponents

Let α, ν, β, p and q be variable exponents. Our aim in this article is to deal with Sobolev embeddings for Riesz potentials of order α with functions f in Morrey spaces L Φ,ν,β(G) with Φ(t) = t p (log(e + t)) q over a bounded open set G ⊂ R n . Here p and q satisfy the log-Hölder and the loglog-Hölder conditions, respectively. Also the case when p attains the value 1 in some parts of the domain is included in our results.

SOBOLEV INEQUALITIES FOR ORLICZ SPACES OF TWO VARIABLE EXPONENTS

Our aim in this paper is to deal with Sobolevs embeddings for Sobolev–Orlicz functions with ∇ u ∈ L p (·) log L q (·) (Ω) for Ω ⊂ n . Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.

Integral Representations, Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on Rn. Our integral representation is an extension of Sobolevs integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.

Continuity and differentiability for weighted Sobolev spaces

Our aim in this paper is to discuss continuity and differentiability of functions in weighted Sobolev spaces in the limiting case of Sobolevs imbedding theorem.

Herz–Morrey spaces of variable exponent, Riesz potential operator and duality

Our aim in this paper is to deal with the boundedness of the Hardy–Littlewood maximal operator on Herz–Morrey spaces and to establish Sobolev’s inequalities for Riesz potentials of functions in Herz–Morrey spaces. Further, we discuss the associate spaces among Herz–Morrey spaces.

Hardy Averaging Operator on Generalized Banach Function Spaces and Duality

Let Af(x) := 1 |B(0,|x|)| ∫ B(0,|x|) f(t) dt be the n-dimensional Hardy averaging operator. It is well known that A is bounded on Lp(Ω) with an open set Ω ⊂ Rn whenever 1 < p ≤ ∞. We improve this result within the framework of generalized Banach function spaces. We in fact find the “source” space SX , which is strictly larger than X, and the “target” space TX , which is strictly smaller than X, under the assumption that the Hardy-Littlewood maximal operator M is bounded from X into X, and prove that A is bounded from SX into TX . We prove optimality results for the action of A and its associate operator A′ on such spaces and present applications of our results to variable Lebesgue spaces Lp(·)(Ω) , as an extension of A. Nekvinda and L. Pick [Math. Nachr. 283 (2010), 262–271; Z. Anal. Anwend. 30 (2011), 435–456] in the case when n = 1 and Ω is a bounded interval.

Boundary behavior of monotone Sobolev functions on John domains in a metric space

This article deals with weighted boundary limits of monotone Sobolev functions on bounded s-John domains in a metric space.

Boundary limits of polyharmonic functions in sobolev-orlicz spaces

This paper deals with weighted boundary limits of polyharmonic functions on a bounded Lipschitz domain G with finite Dirichlet-type integral. There are many known results about the existence of angular limits for harmonic functions. Recently, tangential behaviours are studied mainly for harmonic functions. Our aim in this paper is to study the existence os weighted limit: limk(x)u(x) as x tends to the bounded, the limit is replaced by the usual tangential limit.

Potential theoretic properties of the gradient of a convex function on a functional space

In the previous paper [11], introducing the notions of potentials and of capacity associated with a convex function Φ given on a regular functional space we discussed potential theoretic properties of the gradient ∇Φ which were originally introduced and studied by Calvert [5] for a class of nonlinear monotone operators in Sobolev spaces. For example: (i) The modulus contraction operates. (ii) The principle of lower envelope holds. (iii) The domination principle holds. (iv) The contraction T k onto the real interval [0, k ] ( k > 0) operates. (v) The strong principle of lower envelope holds. (vi) The complete maximum principle holds.

WEIGHTED ORLICZ-RIESZ CAPACITY OF BALLS

Our aim in this note is to estimate the weighted Orlicz-Riesz capacity of balls.