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On the boundary limits of harmonic functions with gradient in L p

Yoshihiro Mizuta — 1984

Annales de l'institut Fourier

This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in L n ( R + n ) , R + n denoting the upper half space of the n -dimensional euclidean space R n . Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta;  Aleš Nekvinda;  Tetsu Shimomura — 2015

Studia Mathematica

Let A α f ( x ) = | B ( 0 , | x | ) | - α / n ∫ B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood maximal operator...

Approximate identities and Young type inequalities in Musielak-Orlicz spaces

Fumi-Yuki Maeda;  Yoshihiro Mizuta;  Takao Ohno;  Tetsu Shimomura — 2013

Czechoslovak Mathematical Journal

We discuss the convergence of approximate identities in Musielak-Orlicz spaces extending the results given by Cruz-Uribe and Fiorenza (2007) and the authors F.-Y. Maeda, Y. Mizuta and T. Ohno (2010). As in these papers, we treat the case where the approximate identity is of potential type and the case where the approximate identity is defined by a function of compact support. We also give a Young type inequality for convolution with respect to the norm in Musielak-Orlicz spaces.

Trudinger's inequality for double phase functionals with variable exponents

Fumi-Yuki Maeda;  Yoshihiro Mizuta;  Takao Ohno;  Tetsu Shimomura — 2021

Czechoslovak Mathematical Journal

Our aim in this paper is to establish Trudinger’s inequality on Musielak-Orlicz-Morrey spaces L Φ , κ ( G ) under conditions on Φ which are essentially weaker than those considered in a former paper. As an application and example, we show Trudinger’s inequality for double phase functionals Φ ( x , t ) = t p ( x ) + a ( x ) t q ( x ) , where p ( · ) and q ( · ) satisfy log-Hölder conditions and a ( · ) is nonnegative, bounded and Hölder continuous.

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