Birman and Solomyak have studied this question quite intensivly.

The paper may not be the easiest to understand, but it does cover in a very general setup, what regularity conditions on the kernel imply in terms of singular number estimates. As you are working with a positive kernel, those are equivalent to the eigenvalues.

If you identify the sphere with a cube $[0,1]^{m-1}$ and transform the measure in a suitable way, you can apply Propositions 2.1-2.3. The technical conditions are a bit more complicated then Hölder continuity, and are defined in section 1.

doi link to the paper

Let $W^{\alpha,p}(X)$ be the fractional Sobolev space with $\alpha$ degrees of differentiability and $p$-integrable functions. Let $W^{\alpha,p}_{hom}(X)$ be the associated homogenuous space (we only care for the highest differential and the function does not need to be $p$-integrable itself).

(2.14): $1- r^{-1} = 2(p^{-1} - \alpha m^{-1})$

To simplify the statement of the theorem
we use the notation $D^\alpha_pX$ in the following sense: for $p\alpha > m$ we take $X = Q^m$ [the unit cube] and $D^\alpha_pX= W^{\alpha,p} Q^m$ ; for $p\alpha < m$ we take $X = \mathbb R^m$ and $D^p_\alpha X =W_{hom}^{\alpha,p}(X)$.
. The notation $D^{\alpha_1}_{p_1}(X) D^{\alpha_2}_{p_2}(Y) $ is to be interpreted similarly (four cases are possible here).

THEOREM 2.5 Let $X = Q^{m_1}$ or $X = \mathbb R^{m_1}$, $Y = Q^{m_2}$ or $Y = \mathbb R^{m_2}$ and $T \in D^{\alpha_1}_{p_1}(X) D^{\alpha_2}_{p_2}(Y)$ for $2 \le p_1$ and $p_2 < \infty$. Let $\rho \in M_{r_1}(X)$ and $\tau \in M_{r_2}(Y)$ with $r_i= 1$ for $p_i\alpha_i > m_i$
and $r_i > 1$ for $ρ_i\alpha_i = m_i$
and suppose that
(2.14) is satisfied when $ρ_i \alpha_ i < m_i$($i=1, 2$). Then
$$s_n(T_{\rho \tau}) \le C n^{-\gamma} N(T \mid D^{\alpha_1}_{p_1}(X)D^{\alpha_2}_{p_2}(Y) ) N^{\frac 1 2} ( \rho \mid M_{r_1}(X) ) N^{\frac 1 2} ( \tau \mid M_{r_2}(Y) ) ,
$$
$$ \gamma = \frac 1 2+ \frac {\alpha_1} {m_1} + \frac {\alpha_2} {m_2}. $$

Here $M_r(X)$ is the space of all measures, such that the density with respect to the Lesbegue measure is in $L^r$. The operator $T_{\rho \tau}$ is the integral operator with kernel $T$ from $L^2(Y,\tau)$ to $L^2(X,\rho)$ and the norm in the composition of spaces $H( X) H'( Y)$ is the space of all kernels, such that the map from $X$ to the Banach space $H'(Y)$ is in the space $H(X)$.

I am not sure how understandable that is. As I said, the paper is not easy to understand and even the statement is quite hard to tell.

Different idea:

Another possible approach would be to consider the Laplace operator on $S^d$ and to show that it satisfies a Weyl law. Then you could use that the operator $(- \Delta)^{-\frac s 2} $ has decaying eigenvalues and is in some $p$ Schatten class (meaning p-summability of the eigenvalues). Than you can conclude that if $(-\Delta)^{\frac s 2 }K$ is in the Hilbert Schmidt class (square integrable), the operator $T$ itself can be written as a comoposition of a $p$-Schatten class operator and a HS operator. Hence, it is in the $Q$-Schatten class for $\frac 1 q= \frac 1 2 + \frac 1 p$ by Hölder.

makes senseyou mean $I_K$ is bounded, then all the eigenvalues are contained in a disk it is not clear to me what you mean by their decay. There are strategies but they are not general in the sense that they require additional assumptions on $K$ so they do not apply to general $K$. Bydefinite kerneldo you meanpositive definite? $\endgroup$1more comment