By Vakhtang Kokilashvili, Alexander Meskhi, Humberto Rafeiro, Stefan Samko
This publication, the results of the authors' lengthy and fruitful collaboration, specializes in essential operators in new, non-standard functionality areas and offers a scientific examine of the boundedness and compactness houses of uncomplicated, harmonic research essential 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 largely utilized to non-linear PDEs, singular integrals and PDO thought. one of many book's such a lot particular beneficial properties is that most of the statements proved listed here are within the type of criteria.
The publication is meant for a large viewers, starting from researchers within the zone to specialists in utilized arithmetic and potential students.
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Additional resources for Integral Operators in Non-Standard Function Spaces: Volume 1: Variable Exponent Lebesgue and Amalgam Spaces
Therefore, ˆ ˆ Ip (M w f ) c + c |f (y)|p1 (y)·p− dy = c + c |f (y)|p(y) dy Ω Ω which completes the proof in this case. n M (w) < p (x : reduction to the case of power weights. 39), we may split integration over Ω into two parts, one over a small neighbourhood B(x0 , δ) of the point x0 , and another over its exterior Ω\B(x0 , δ), and to choose δ suﬃciently small so that the 0 ,δ))−1 0 )−1 1 number p− (B(x is arbitrarily close to p(x p(x0 ) p(x0 ) = p (x0 ) . 42) =: M1w + M2w + M3w + M4w . Since the weight is strictly positive and bounded beyond any neighbourhood of the point x0 , we get M4w f (x) CM f (x).
13. 37). Then 1 Lr(·) (Rn ) e ς 0 Ap . 2. Convolution Operators 13 Proof. 11) for every m > 0 there holds the (e + |x|)−m with γ e−mAp . Taking m > n and λ = emAp we estimate γ r(x) then have ˆ ˆ r(x) 1 dx dx =: Cm . 39). en−m (m−1)(m−2)···(m−n) . 3 Estimation of the Norm of Convolution Operators ˆ Let k(x − y)f (y) dy Kf (x) := Rn be a convolution operator. 14 are deﬁned by 1 1 1 + , =1− r0 p(∞) q(∞) 1 1 1 =1− + , s0 p− q+ 1 and s0 r0 1 used in s0 . 14. Let p, q ∈ P∞ (Rn ) and q(∞) . p(∞). 38), and 2 case p− = p∞ .
24. Let Ω ⊂ Rn be a bounded open set and p ∈ Plog (Ω). 28), if n r p(xk ) wk (r) ∈ Φ0n , k = 1, 2, . . , N, or, equivalently, if the MO indices of the functions wk (r) satisfy the conditions − n < m(wk ) p(xk ) M (wk ) < Proof. We have to show that M w f equivalent to Ip (M w f ) c for f n , k = 1, 2, . . , N. p (xk ) c in some ball f p(·) Ip (M f ) p(·) f (y) w(|y − x0 |) w(|x − x0 |)p(x0 ) M c Ω We ﬁrst prove the bound M w f − where 1 q0 = p− −1 p(x0 ) . p(·) n < m(w) p(x0 ) Observe that 1 q0 R, which is p(·) R.