Rotating Oblique Differences and Fractional Sobolev Spaces

We introduce a new class of rotating oblique differences: for \( h\in\mathbb{R} \) and a direction \( e_i\in\mathbb{R}^k \), we define\( \Delta_{h,i}^{\Omega}f(x) := f(x+he_i) - f(x+hL_\Omega e_i), \) where \( L_\Omega \) is the skew-symmetric generator of the rotation. This definition samples \( f \) along the oblique direction \( e_i - L_\Omega e_i \), combining anisotropy, fractional order, and rotation inside the difference operator rather than as external corrections. The associated rotating oblique Gagliardo seminorm generates the rotating fractional Sobolev space \( \mathcal{W}^{\nu,p}_{\alpha,\Omega}(\mathbb{R}^k) \). We prove unconditionally that this space is a Banach space, is reflexive for \( 1 < p < \infty \), and supports a one-sided Gagliardo--Besov comparison with explicit polynomial dependence on \( (1+|\Omega|) \) in the constants. Our central result is the exact Fourier characterisation of the seminorm for \( p=2 \):\( [f]^2_{\mathcal{W}_i^{\nu/\alpha_i,2,\Omega}} = \Lambda_{\nu,\alpha_i}\int_{\mathbb{R}^k} \bigl|\xi_i - (L_\Omega e_i)\cdot\xi\bigr|^{2\nu/\alpha_i} |\hat f(\xi)|^2\,d\xi, \) where the multiplier is a linear oblique form, not a coordinate power nor a radial power. The multiplier degenerates on the hyperplane \( \omega_i(\xi) = 0 \), and we characterise precisely how this degeneracy controls the constants in the one-sided comparison; the reverse inequality, and hence the full Gagliardo--Besov equivalence, remains conjectural. We complement these results with an anisotropic Hardy--Littlewood maximal inequality whose constants do not depend on the rotation rate (the operator itself is \( \Omega \)-independent), and with a vanishing-mean identity for the first-order Baker--Campbell--Hausdorff correction under the symmetrised anisotropic measure with a reflection-symmetric cutoff. The Sobolev embedding, the first- and higher-order rotating Landau inequalities, and the effective-dimension identity are stated as explicitly labelled conjectures, with no claim of proof; the unconditional core is the framework, the exact Fourier characterisation, the one-sided comparison, the maximal inequality, and the vanishing-mean identity.