I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as follows (for some fixed $i$ and $j$)

$$ F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) \right)f(y)dy $$

where $f \in C_c^{\gamma}(\mathbb{R}^n)$ (ie a compactly supported $\gamma$-Hölder continuous function) and the integral kernel $k:\mathbb{R}^n\backslash \lbrace \vec{0} \rbrace \rightarrow \mathbb{R}^n$ is defined as

\begin{equation} k_i(x) =\frac{x_i}{|x|^n} \end{equation}

Note $k$ is the derivative of the Newtonian kernel up to a constant factor. In addition, we have $Y: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that

$$ \| \nabla Y \|_{1, \gamma} = |Y(0)|+\|\nabla Y\|_0 + |\nabla Y|_{1,\gamma} < \infty $$ where $$ |W|_{\gamma} = \sup_{x \neq y} \frac{|W(x)-W(y)|}{|x-y|^{\gamma}} $$ is the $\gamma$-Hölder seminorm. Note $Y$ has bounded $\gamma$-Hölder continuous derivatives.

-----------------------------Attempt at Solution -------------------------------------

In attempting to prove the $\gamma$-Holder continuity of $F$, my strategy thus far has involved expanding $Y$ as a first order Taylor polynomial as follows $$ Y_k(y)=Y_k(x) + \partial_l Y_k(x)(y_l-x_l) +(y_l-x_l)\int_0^1 \left( \partial_l Y_k(x+t(y-x)) -\partial_l Y_k(x) \right)dt $$ I'll denote the remainder term as $R_x(y)$. Note $|R_{x}(y)| \leq C|y-x|^{1+\gamma}$.

So I write \begin{eqnarray} && F(x+h)-F(x)=\lim_{\epsilon \rightarrow 0} \bigg( \int_{|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)f(y)dy \\ &&- \ \int_{|x-y|\geq \epsilon} \partial_k \partial_j k_i(x-y)\partial_lY(x)(x_l+h_l-y_l)f(y)dy -\int_{|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lR_{x+h}(y)f(y)dy \\ && + \int_{|x-y|\geq \epsilon} \partial_k \partial_j k_i(x-y)\partial_lR_{x}(y)f(y)dy \bigg) \end{eqnarray}

I can show the absolute value of the difference of the first two terms is bounded by $C|h|^{\gamma}$. This is done by first observing that $\partial_j k_i$ is the integral kernel of a singular integral operator (ie it is smooth everywhere but $\vec{0}$, is homogeneous of degree $-n$, and has mean value on the unit sphere). In fact, this mean-value property extends as follows:

\begin{equation} \int_{|x|=1}x^{\beta}D^{\alpha}k_i(x)dS=0 \quad \Rightarrow \int_{r \leq |x|\leq R} x^{\beta}D^{\alpha}k_i(x) dx = 0 \end{equation}

for any multi-indices $\alpha$ and $\beta$ such that $|\alpha|=|\beta|+1$. In particular terms of the form

\begin{equation} C\int_{r \leq |x|\leq R} x_l \partial_k\partial_jk_i(x) dx = 0 \end{equation} for C constant can be added freely.

The first two terms can be dealt with (roughly, this is jut a sketch) in two steps. The first is

\begin{eqnarray} && \int_{ |h|>|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)f(y)dy \\ && = \int_{ |h|>|x+h-y|\geq \epsilon} \partial_k \partial_j k_i(x+h-y)\partial_lY(x+h)(x_l+h_l-y_l)\left(f(y)-f(x) \right) dy \leq c|f|_{\gamma}|h|^{\gamma} \end{eqnarray}

where we made use of the $\gamma$-Holder continuity of $f$ and the mean-value property of the kernel by adding a zero term. The second is

\begin{eqnarray} && \int_{ |x+h-y|\geq|h|,|x-y|\geq 2|h|} \Big( \partial_k \partial_j k_i(x+h-y)(x_l+h_l-y_l) - \partial_k \partial_j k_i(x-y)(x_l-y_l) \Big)\left(f(y)-f(x+h) \right) dy \\ && \leq c|f|_{\gamma}|h|^{\gamma} \end{eqnarray}

through the use of the mean-value theorem on the integral kernel $\partial_k \partial_j k_i$ and the further addition of zero terms.

--------------- Remaining Problems -----------------

I have trouble bounding the difference of the remainder terms by $C|h|^{\gamma}$. Following a similar method as above I can bound them by $C|h|^{\xi}$ for $\xi < \gamma$. But perhaps there is a more general theorem I could use? I'd also be interested in textbook recommendations covering similar subject matter.


1 Answer 1


This is covered in Gilbarg-Trudinger, and I think it can be put under the heading "potential theory approach to Schauder estimates". "Kellog's theorem" might reveal something too. It is closely related to the Calderon-Zygmund theory of singular integral operators, although the latter is typically concerned with $L^p$ spaces.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.