By Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko

ISBN-10: 3319210149

ISBN-13: 9783319210148

ISBN-10: 3319210157

ISBN-13: 9783319210155

This e-book, the results of the authors' lengthy and fruitful collaboration, specializes in vital operators in new, non-standard functionality areas and provides a scientific examine of the boundedness and compactness homes of uncomplicated, harmonic research vital operators within the following functionality areas, between others: variable exponent Lebesgue and amalgam areas, variable Hölder areas, variable exponent Campanato, Morrey and Herz areas, Iwaniec-Sbordone (grand Lebesgue) areas, grand variable exponent Lebesgue areas unifying the 2 areas pointed out above, grand Morrey areas, generalized grand Morrey areas, and weighted analogues of a few of them.

The effects acquired are generally utilized to non-linear PDEs, singular integrals and PDO concept. one of many book's such a lot targeted positive factors is that most of the statements proved listed here are within the type of criteria.

The e-book is meant for a vast viewers, starting from researchers within the region to specialists in utilized arithmetic and potential students.

**Read Online or Download Integral Operators in Non-Standard Function Spaces: Volume 1: Variable Exponent Lebesgue and Amalgam Spaces PDF**

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**Extra info for Integral Operators in Non-Standard Function Spaces: Volume 1: Variable Exponent Lebesgue and Amalgam Spaces**

**Sample text**

2 The Case of Variable p In the case of variable exponent p(x), on R+ , we will suppose that f ∈ P0,∞ (R+ ) and impose the decay conditions at the origin and inﬁnity of the form |p(x) − p0 | 1 , 2 C , 0

38), and 2 case p− = p∞ . 1 1+ p2 + p∞ − 2 may be replaced by 2 p− + p1∞ in the 14 Chapter 1. Hardy-type Operators in Variable Exponent Lebesgue Spaces Proof. 40), deﬁne 1 1 1 + =1− . s1 p(∞) q+ 1 1 1 , =1− + r1 p− q(∞) Then 1 min{r1 , s1 } r0 max{r1 , s1 } ∞. s0 By the classical Young inequality for the convolution operator K, we have Kf q+ k s0 f p− , Kf q(∞) k r1 f p− , Kf q+ k s1 f p(∞) , Kf q(∞) k r0 f p(∞) . 44) Therefore, Kf q+ k Ls0 ∩Ls1 f Lp− +Lp(∞) q(∞) k Lr0 ∩Lr1 f Lp− +Lp(∞) . 45) is a consequence of the continuous embeddings Lr0 ∩ Ls0 → Lr1 ∩ Ls0 , Lr0 ∩ Ls0 → Lr0 ∩ Ls1 with the norm of the embedding operator equal to 1.

58) +) hold if and only if α and β satisfy the conditions α(0) < 1 , p (0) α(∞) < 1 , p (∞) respectively β(0) > − 1 , p(0) β(∞) > − 1 . 59) Proof. A) Suﬃciency. 1◦ . The case where p(0) = p(∞), μ(0) = μ(∞), α(0) = α(∞) and β(0) = β(∞). In this case, by the decay condition we have the equivalence xμ(x) ≈ xμ(0) , xα(x) ≈ xα(0) , xβ(x) ≈ xβ(0) , on the whole half-line R+ , so that our Hardy operators H α,μ , Hβ,μ are equivalent to the Hardy operators with constant exponents μ = μ(0), α = α(0), β = β(0), respectively.

### Integral Operators in Non-Standard Function Spaces: Volume 1: Variable Exponent Lebesgue and Amalgam Spaces by Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko

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