Miroslav Krbec

On Decomposition in Exponential Orlicz Spaces

Various characterizations are given of the exponential Orlicz space L and the Orlicz-Lorentz space L. By way of application we give a simple proof of the celebrated theorem of Brezis and Wainger concerning a limiting case of a Sobolev imbedding theorem.

Conical differentiability for evolution variational inequalities

Abstract The conical differentiability of solutions to the parabolic variational inequality with respect to the right-hand side is proved in the paper. From one side the result is based on the Lipschitz continuity in H 1 2 ,1 (Q) of solutions to the variational inequality with respect to the right-hand side. On the other side, in view of the polyhedricity of the convex cone K={v∈ H ;v |Σ c ⩾0,v |Σ d =0}, we prove new results on sensitivity analysis of parabolic variational inequalities. Therefore, we have a positive answer to the question raised by Fulbert Mignot (J. Funct. Anal. 22 (1976) 25–32).

On extrapolation blowups in the

Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operatorOpen image in new window acting continuously inOpen image in new window forOpen image in new window close toOpen image in new window and/or takingOpen image in new window intoOpen image in new window asOpen image in new window and/orOpen image in new window with norms blowing up at speedOpen image in new window and/orOpen image in new window,Open image in new window. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens ifOpen image in new window asOpen image in new window. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs forOpen image in new window . We also touch the problem of comparison of results in various scales of spaces.Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space Lω(w). For a Young function Ω ∉ Δ2(∞) we present a necessary and sufficient conditions in order that Lω(w) = Lω(XΩ) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from Lω into itself when ω Δ2(∞) and obtain necessary and sufficient condition in order that the composition operator maps Lω. continuously onto Lω.

Imbeddings of Brezis-Wainger type. The case of missing derivatives

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

On extrapolation blowups in the scale

Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operatorOpen image in new window acting continuously inOpen image in new window forOpen image in new window close toOpen image in new window and/or takingOpen image in new window intoOpen image in new window asOpen image in new window and/orOpen image in new window with norms blowing up at speedOpen image in new window and/orOpen image in new window,Open image in new window. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens ifOpen image in new window asOpen image in new window. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs forOpen image in new window . We also touch the problem of comparison of results in various scales of spaces.Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.

Moduli and Characteristics of Monotonicity in Some Banach Lattices

First the characteristic of monotonicity of any Banach lattice is expressed in terms of the left limit of the modulus of monotonicity of at the point . It is also shown that for Köthe spaces the classical characteristic of monotonicity is the same as the characteristic of monotonicity corresponding to another modulus of monotonicity . The characteristic of monotonicity of Orlicz function spaces and Orlicz sequence spaces equipped with the Luxemburg norm are calculated. In the first case the characteristic is expressed in terms of the generating Orlicz function only, but in the sequence case the formula is not so direct. Three examples show why in the sequence case so direct formula is rather impossible. Some other auxiliary and complemented results are also presented. By the results of Betiuk-Pilarska and Prus (2008) which establish that Banach lattices with and weak orthogonality property have the weak fixed point property, our results are related to the fixed point theory (Kirk and Sims (2001)).

On solutions to the heat equation with the initial condition in the Orlicz—Slobodetskii space

We study the boundary-value problem ~t = x~ u(x;t), ~ u(x; 0) = u(x), where x2 ;t2 (0;T ), R n 1 is a bounded Lipschitz boundary domain, u belongs to certain Orlicz-Slobodetskii space Y R;R (). Under certain assumptions on the Orlicz function R, we prove that the solution u belongs to Orlicz-Sobolev space W 1;R ( (0;T )).

A Note on Noneffective Weights in Variable Lebesgue Spaces

We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we show that L p ( ⋅ ) ( Ω ) = L ω p ( ⋅ ) ( Ω ) if and only if ω ( x ) 1 / p ( x ) ~ constant in the set where p ( ⋅ ) ∞ , and ω ( x ) ~ constant in the set where p ( ⋅ ) = ∞ .

On extrapolation blowups in the Open image in new window scale

Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operatorOpen image in new window acting continuously inOpen image in new window forOpen image in new window close toOpen image in new window and/or takingOpen image in new window intoOpen image in new window asOpen image in new window and/orOpen image in new window with norms blowing up at speedOpen image in new window and/orOpen image in new window,Open image in new window. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens ifOpen image in new window asOpen image in new window. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs forOpen image in new window . We also touch the problem of comparison of results in various scales of spaces.Yanos extrapolation theorem dated back to 1951 establishes boundedness properties of a subadditive operator acting continuously in for close to and/or taking into as and/or with norms blowing up at speed and/or,. Here we give answers in terms of Zygmund, Lorentz-Zygmund and small Lebesgue spaces to what happens if as. The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for . We also touch the problem of comparison of results in various scales of spaces.