1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

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2mean a function u(t), t ∈ R, taking values in the set of generalized functions on R dsuch that, for any t, s ∈ R satisfying t ≥ s and test function ϕ ∈ C ∞ 0 (R d ),wehave(u(t),ϕ)=(u(s),ϕ)+∫ ts[a ij (r)(u(r),ϕ xi xj)+(f(r),ϕ)] dr.The emphasis is on proving solvability in function spaces of Sobolev type with differentpowers of summability p and q with respect to x and t. This issue arose from the theoryof stochastic partial differential equations in domains in Sobolev spaces with weightsand it turns out that, in this theory, the spaces with weights are the only reasonableones where to look for solutions to equations in domains. To illustrate this necessityconsideru t = u xx in R 2 + with u(0,x)=0,u(t, 0) = g(t),where g is a bounded nondifferentiable function. Thenu xx cannot be continuous sinceotherwise0=u(t, 0) − u(0, 0) = g(t) − g(0) =∫ t0u xx (s, 0) ds.This equation arrises from the SPDEu(t, x) − u(0,x)=∫ t0u xx (s, x) ds + w t ,where w t is a one-dimensional Wiener process. Hence comes the need of weights.
Next, equation (1.1) will be considered in L q ((0,T),L p )-spaces, so that the powersof summability with respect to x and t are different.Surprisingly enough, to the best of my knowledge, the case q ≠ p was never addressed3before even for the heat equation in R × R d without weights.We could only findreferences [1] and [12], where different powers of summability can be related to theCauchy problem in {t >0} for f =0. It turns out that in R × R d the result we needcan be obtained quite easily on the basis of a Banach space version of the Calderón-Zygmund theorem (see Sec. 2), which allows one to pass from q = p to q ≠ p.For equation (1.1) in R×R d + we impose zero boundary condition and look for solutionsin weighted Sobolev spaces with weights allowing the spatial derivatives of solutions toblow up near the boundary x 1 =0. Our results in this setting extend the correspondingresults in [5], where q = p.This time we again use the Calderón-Zygmund theorem starting with the results validfor q = p. However, in order to check the conditions of this theorem, we need somenontrivial properties of the heat semigroup in weighted spaces, which we prove in Sec. 4.In Sec. 5, this allow us to get the result for R×R d +, but only if a 1j ≡ 0 for j ≥ 2. Usually,if one proves a sufficiently strong result for the heat equation, the same or very close
Page 1: THE HEAT EQUATION IN L q ((0,T),L p
Page 5 and 6: Theorem 1.1. Let p, q ∈ (1, ∞),
Page 7 and 8: 7Also remember that, for f ∈ C
Page 9 and 10: where K(x, y) is a bounded operator
Page 11 and 12: In order to prove that the assumpti
Page 13 and 14: where LV = a rs (V xr1 x s 1 + ...
Page 15 and 16: Definition 3.1. Take and fix a nonn
Page 17 and 18: Theorem 3.3. (i) We have17(H γ p,
Page 19: 194. Some smoothing properties of t
Page 23 and 24: Now we claim that Ā is a bounded o
Page 25 and 26: Proof. Clearly (6.1) becomes strong
Page 27 and 28: 27≤ N(||Mf(t)|| p H γ−2p,θ+ |

1. Introduction We are going to investigate the equation ut(t, x ... - ICMS

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