Is there any estimate available for the derivatives of the fractional heat kernel? Estimates on the kernel itself are at this link.
Also is any estimate available if we consider the problem with homogeneous Dirichlet boundary conditions?
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$\begingroup$ Some etimates may be found following to J.Kemppainen, J.Siljander, V.Verrgara, R. Zacher Decay estimates for time-fractional and other non-local in time subdiffusion equations in $Rd$ Math. Ann. (2016) 366:941–979. $\endgroup$– user171871Commented Jan 27, 2021 at 10:58
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$\begingroup$ @Logiyko: These references seem to deal with time-fractional models, where the derivative w.r.t. time is raised to a fractional power. Jay's question seems to be related to space-fractional models, where the spatial derivatives (the Laplace operator) is raised to a fractional power. $\endgroup$– Mateusz KwaśnickiCommented Jan 27, 2021 at 23:35
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$\begingroup$ I'm sorry. The correct reference is J.Kemppainen, J.Siljander, V.Verrgara, R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, JDE V.263,pp.149-201, 2017. One should take $\alpha=1$, $u_0=0$ and consider only $Y(x,t)$ $\endgroup$– user171871Commented Jan 28, 2021 at 7:06
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In full space: sure they are! The simplest way to get to them is to observe that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) , \tag{A} $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (x_1, \ldots, x_n, 0, 0) \in \mathbb R^{n+2}$, and $p_t^{(n)}(x)$ is the heat kernel for $(-\Delta)^s$ in $n$ dimensions. The above observation is a direct application of the following Bochner's relation for the radial function $f(\xi) = \exp(-t |\xi|^{2s})$, $V(x) = x_j$, and $\ell = 1$.
Bochner's relation (see Corollary on page 72 in [Stein]) Let $f$ and $g$ be two radial Schwartz functions in $\mathbb R^n$ and $\mathbb R^{n+2\ell}$, with the same profile function (i.e. $f(\xi) = g(\tilde \xi)$ if $|\xi| = |\tilde \xi|$), and let $V$ be a homogeneous harmonic polynomial of degree $\ell$. Suppose that $x \in \mathbb R^d$, $\tilde x \in \mathbb R^{d + 2 \ell}$ satisfy $|x| = |\tilde x|$. Then $$ \mathscr F_n [-i^\ell V(\xi) f(\xi)](x) = V(x) \mathscr F_{n+2\ell} \tilde f(\tilde{x}) . $$ Here $\mathscr F_n$ denotes the $n$-dimensional Fourier transform.
Formula (A) is stated explicitly in Theorem 1.5 in [KR] (in a greater generality; I bet there is an earlier reference for the fractional Laplacian).
For the Dirichlet heat kernel, there is a whole line of related research by various authors (Bogdan, Chen, Grzywny, Jakubowski, Kim, Kulczycki, Ryznar, Song, Szczypkowski, Vondraček). The upper bound $$ |\nabla_x p_t^D(x,y)| \leqslant \frac{C}{\min\{\operatorname{dist}(x,D^c), t^{1/(2s)}\}} p_D(t, x, y) $$ for the gradient of the heat kernel of $(-\Delta)^s$ in an open set $D$ is given as Example 5.1 in [KR] (note: this may have been written in some earlier work). And the estimates of $p_D(t, x, y)$ are well-known: they were established in [CKS] for domains with smooth ($C^{1,1}$) boundary, and in [BGR] for Lipschitz sets.
For more general "boundary" conditions, the answer depends on what kind of boundary condition you are interested in, but I suppose this has not been studied in a great generality.
References:
[BGR] K. Bogdan, T. Grzywny, M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 38(5) (2010): 1901–1923.
[CKS] Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimates for Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12 (2010): 1307–1329.
[KR] T. Kulczycki, M. Ryznar, Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Amer. Math. Soc. 368 (2016), no. 1, 281-318.
[Stein] Stein, E.M., Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
answered Jan 27, 2021 at 10:50
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$\begingroup$ Thank you. In a bounded domain, I mean to consider the fractional Laplacian as in the link and the homogeneous Dirichlet condition in $\mathbb R^n \setminus \Omega$ (not the semigroup fractional Laplacian, which seems to be the one in the papers you mentioned). Do you know any reference on that? $\endgroup$– JayCommented Jan 27, 2021 at 22:16
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$\begingroup$ I added a reference. I did not understand the part of your comment about the semigroup fractional Laplacian — we seem to be talking about the same object. $\endgroup$ Commented Jan 27, 2021 at 23:19
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$\begingroup$ I'm talking about the fractional Laplacian defined as a singular integral with the "boundary condition" on $\mathbb R^n \setminus \Omega$ $\endgroup$– JayCommented Jan 30, 2021 at 10:09
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$\begingroup$ Thank you. Now I realized that they indeed consider the same operator. I have just a related question: In the whole space, from the link we see that $p_t^{n+2}(\tilde x) = t^{-(n+2)/2s}k(|\tilde x|/t^{-1/(2s)}))$. What are the properties of $k$. In particular, is $(k_s(r))_{t>0} := (k(|\tilde x|/t^{-1/(2s)}))_{t>0}$ dense in $C_0(0,\infty)$? $\endgroup$– JayCommented Jan 30, 2021 at 13:12
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$\begingroup$ How could it be dense? Did you mean "linearly dense"? In any case, it is not even in $C_0((0,\infty))$, as it does not vanish at zero. $\endgroup$ Commented Jan 30, 2021 at 16:06