Weighted Two-parameter Bergman Space Inequalities
نویسندگان
چکیده
In this inequality, ∇ denotes the full gradient in R + : ∇ = (∂/∂x1, . . . , ∂/∂xd, ∂/∂y); R + is the usual upper half space Rd×(0,∞); μ is a positive Borel measure defined on R + ; and v is a non-negative function in Lloc(R d). We studied this inequality primarily for p and q in the range 1 < p ≤ q < ∞. For the case in which q ≥ 2, we proved sufficient conditions on μ and v (depending on p, q, and d) for the inequality (1.1) to hold for all f ∈ ∪1≤r<∞L(R, dx). The argument in [WhWi] began with the observation that (1.1) is a special case of a more general inequality. Let h be a smooth function with decay at infinity (precisely how much decay will be specified later), defined on Rd. For y > 0, set hy(x) = y−dh(x/y), the usual L1-dilation. If we set u(x, y) = f ∗hy(x), then any component of ∇u(x, y) can be written as f ∗ (yφy)(x), where φ is smooth, has some decay, and in addition satisfies ∫
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تاریخ انتشار 2002