By Alexander Meskhi

This booklet is dedicated to the degree of non-compactness (essential norm) in weighted Lebesgue areas for maximal, capability and singular operators dened, more often than not conversing, on homogeneous teams. the most issues of the monograph comprise similar effects for capability and singular integrals in weighted functionality areas with non-standard development. one of many major attribute gains of the monograph is that the issues are studied within the two-weighted atmosphere and canopy the case of non-linear maps, corresponding to, Hardy-Littlewood and fractional maximal services. sooner than, those difficulties have been investigated just for the constrained type of kernel operators consisting in simple terms of Hardy-type and Riemann-Liouville transforms. The ebook will be regarded as a scientific and targeted research of a category of particular fundamental operators from the boundedness/compactness or non-compactness standpoint. the fabric is self-contained and will be learn by means of people with a few history in genuine and useful research.

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**Example text**

Then M K (Lwp (Rn )) ≥ max{ sup lim I(a, τ);CB,n ¯ sup lim I(a, τ)}, n τ→∞ n τ→0 a∈R where I(a, τ) := |B(a, τ)| a∈R 1/p −1 1−p′ w(x)dx B(a,τ) ¯ and n. and CB,n ¯ is a constant depending only on B w B(a,τ) 1/p′ (x)dx 30 Alexander Meskhi Proof. 4 it suffices to show that w(B(a, η1 τ) \ B(a, τ)) ≥ cw(B(a, τ)), where w(E) := E w and the positive constant c depends only on B¯ and n. But it is easy to verify that if w ∈ A p (Rn ), then ¯ w(B(a, 2τ)) ≤ cn Bw(B(a, τ)) for all τ > 0 and a ∈ Rn . 1 we have w(B(a, η1 τ)) \ B(a, τ)) ≥ (η2 − 1)w(B(a, τ)), ¯ where η2 depends only on n and B.

6. Let I = [0, a] (0 < a < ∞) and let 1 ≤ p− (I) ≤ p0 (x) ≤ q(x) ≤ q+ (I) < ∞ for almost every x ∈ I. Then the condition a sup 0

2) yields cB( j) ≤ λ Kernel Operators on Cones 49 for every integer j, j ≤ n. Hence sup j≤n B( j) ≤ cλ for all integers n with the condition 2n < α. Therefore lim sup j≤n B( j) ≤ cλ. n→−∞ Now we take m ∈ Z such that 2m > β. Then for f j (y) = χE2 j+1 ) (y) ( j ≥ m), we obtain kq (x, δ1/(2c0 ) x)v(x)r(x)q dx. |K f j (x)|q v(x)dx ≥ c E2 j+1 \E2 j E2 j+1 \E2 j q On the other hand, f j X = c2Q jq/p . Hence sup j≥m B( j) ≤ cλ, where c depends only on p, q and c1 . Consequently, lim sup j≥m B( j) ≤ cλ from which it follows the desired estimate.