By Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko
ISBN-10: 3319210173
ISBN-13: 9783319210179
ISBN-10: 3319210181
ISBN-13: 9783319210186
This ebook, the results of the authors’ lengthy and fruitful collaboration, makes a speciality of necessary operators in new, non-standard functionality areas and provides a scientific examine of the boundedness and compactness houses of easy, harmonic research imperative 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 greatly utilized to non-linear PDEs, singular integrals and PDO idea. one of many book’s so much targeted positive aspects is that almost all of the statements proved listed here are within the type of criteria.
The booklet is meant for a large viewers, starting from researchers within the sector to specialists in utilized arithmetic and potential students.
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Extra resources for Integral Operators in Non-Standard Function Spaces: Volume 2: Variable Exponent Hölder, Morrey–Campanato and Grand Spaces
Sample text
1. Let 1 L p,λ f∈ (Ω) = Lploc (Ω) 0. The Morrey space Lp,λ (Ω) is defined as ˆ 1 : sup λ r x∈Ω;r>0 |f (y)|p dy < ∞ . 1) B(x,r) This is a Banach space with respect to the norm f Lp,λ (Ω) := sup x∈Ω;r>0 ˆ 1 rλ 1 p |f (y)| dy p . 2) B(x,r) Note that if we want to have an equivalent norm in the form ˆ 1 sup 1 p |f (y)| dy p λ |B(x, r)| n x∈Ω;r>0 , B(x,r) we should assume that Ω satisfies the condition |B(x, r)| Crn , 0 < r < diam (Ω), which is done, for instance, in the book of Kufner, John, and Fuˇc´ık [151].
54) be satisfied. 49), the operator I α(·) is bounded from the space H ω(·) (Ω) to the weighted space H ωα (·) (Ω, α). Proof. 19. 28. 55) reduces to sup [λ(x) + Re α(x)] < θ. 3. Potentials of Constant Order 593 In the following theorem we use the notation ω−α (x, t) = t− Re α(x) ω(x, t) and ω−α (x, h) = sup ω−α (y, h) y:|y−x| 2]. 18. Let ω(x, t) be in ∈ W (0, ) uniformly in x. Then ω(x, t) ∈ Zβ ⇐⇒ M (w) < β. 17. 19. Let ωδ = ω(x,t) tδ(x) and ωβ = ω(x,t) . 36) ω(x, t) ∈ Zβ(·) ⇐⇒ M (wβ ) < 0. 37) take the form ω(x, t) ∈ Zδ(·) ⇐⇒ ω(x, t) ∈ Zβ(·) ⇐⇒ inf [m(w, x) − δ(x)] > 0, x∈Ω sup[M (w, x) − β(x)] < 0. x∈Ω We will make use of the following property of the bounds for functions ω(x, t) ∈ W(T) in terms of their indices: c1 tM(w)+ε ω(x, t) c2 tm(ω)−ε , 0 t (< ∞), where ε > 0 and the constants c1 , c2 may depend on ε, but do not depend on x and t (see Samko [192, Thm.