Carleson Measure Problems for Parabolic Bergman Spaces and Homogeneous Sobolev Spaces
نویسنده
چکیده
Let bα(R 1+n + ) be the space of solutions to the parabolic equation ∂tu+ (−△)u = 0 (α ∈ (0, 1]) having finite L(R 1+n + ) norm. We characterize nonnegative Radon measures μ on R + having the property ‖u‖Lq(R1+n + ,μ) . ‖u‖ Ẇ1,p(R + ) , 1 ≤ p ≤ q < ∞, whenever u(t, x) ∈ bα(R 1+n + ) ∩ Ẇ 1.p(R + ). Meanwhile, denoting by v(t, x) the solution of the above equation with Cauchy data v0(x), we characterize nonnegative Radon measures μ on R 1+n + satisfying ‖v(t2α , x)‖ Lq(R + ,μ) . ‖v0‖Ẇβ,p(Rn), β ∈ (0, n), p ∈ [1, n/β], q ∈ (0,∞). Moreover, we obtain the decay of v(t, x), an iso−capacitary inequality and a trace inequality.
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