Estimates of fractional heat kernel

• 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?

• 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.
– user171871
Jan 27, 2021 at 10:58
• @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. Jan 27, 2021 at 23:35
• 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 $Y(x,t)$
– user171871
Jan 28, 2021 at 7:06

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)

• 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?
– Jay
Jan 27, 2021 at 22:16
• 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. Jan 27, 2021 at 23:19
• I'm talking about the fractional Laplacian defined as a singular integral with the "boundary condition" on $\mathbb R^n \setminus \Omega$
– Jay
Jan 30, 2021 at 10:09
• 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 $C_0(0,\infty)$?
– Jay
Jan 30, 2021 at 13:12
• 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. Jan 30, 2021 at 16:06