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The form boundedness criterion for the relativistic Schrödinger operator

Vladimir Maz'ya, Igor Verbitsky (2004)

Annales de l’institut Fourier

We establish necessary and sufficient conditions on the real- or complex-valued potential Q defined on n for the relativistic Schrödinger operator - Δ + Q to be bounded as an operator from the Sobolev space W 2 1 / 2 ( n ) to its dual W 2 - 1 / 2 ( n ) .

The homogeneous transfinite diameter of a compact subset of N

Mieczysław Jędrzejowski (1991)

Annales Polonici Mathematici

Let K be a compact subset of N . A sequence of nonnegative numbers defined by means of extremal points of K with respect to homogeneous polynomials is proved to be convergent. Its limit is called the homogeneous transfinite diameter of K. A few properties of this diameter are given and its value for some compact subsets of N is computed.

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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